Exact Functional ANOVA Decomposition for Categorical Inputs Models

2026-03-03

AI Summary

This work addresses the limitations of existing methods in handling categorical inputs with arbitrary dependency structures, which often lack explicit closed-form expressions and rely on computationally expensive sampling approximations. By integrating functional analysis with discrete Fourier analysis, the paper presents the first distribution-free, exact functional ANOVA decomposition framework for categorical variables under arbitrary dependencies. This approach naturally extends SHAP values to general categorical settings and accommodates non-rectangular support distributions. It not only subsumes the independent case as a special instance but also efficiently captures complex dependency structures, thereby significantly enhancing both the computational efficiency and applicability of model interpretability techniques in realistic scenarios.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Graphical Models
• Computer Vision: Interpretability, Explainability, and Transparency

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Functional ANOVA offers a principled framework for interpretability by decomposing a model's prediction into main effects and higher-order interactions. For independent features, this decomposition is well-defined, strongly linked with SHAP values, and serves as a cornerstone of additive explainability. However, the lack of an explicit closed-form expression for general dependent distributions has forced practitioners to rely on costly sampling-based approximations. We completely resolve this limitation for categorical inputs. By bridging functional analysis with the extension of discrete Fourier analysis, we derive a closed-form decomposition without any assumption. Our formulation is computationally very efficient. It seamlessly recovers the classical independent case and extends to arbitrary dependence structures, including distributions with non-rectangular support. Furthermore, leveraging the intrinsic link between SHAP and ANOVA under independence, our framework yields a natural generalization of SHAP values for the general categorical setting.

Problem

• Functional ANOVA
• categorical inputs
• dependent features
• SHAP values
• closed-form decomposition

Innovation

• Functional ANOVA
• categorical inputs
• closed-form decomposition
• SHAP generalization
• discrete Fourier analysis